It does not happen on Windows XP with Office 2003.  The problem occurrs on
Windows 2000 with Office 2000 and all of the available updates for each.  
Security updates KB912919 and KB896424 cause this behavior to appear.

Here are the steps to cause the problem to appear:

   - Inside the new blank document go to the "Insert" menu.
   - Switch to Page Layout view.
   - Click "Object" to insert an object.
   - Select "Image Document" (or "Calendar Control 9.0" or maybe some
others) from the list.
   - If you selected "Image Document", draw something inside the box.
   - Select something outside of the newly inserted object, maybe type some
text.
   - Save the document as RTF (Rich Text Format).
   - Notice the vanishing act.

Also, if you want to, you can set up a further informative test.  
   - Open another program and position it so it covers about half of the
DNO or other object inside the Word document.
   - Switch back to Word and save the document, causing the objects to
vanish.
   - Alt+Tab to the other program.
   - Alt+Tab back to Word.
   - Notice that the objects are only half drawn.
 
I strongly suspect that an update is needed for Word, but do not know where
else to post this question.